The Standard Form of a Quadratic Function:
[tex]\text{ y = ax}^2\text{ + }bx\text{ + c}[/tex]Using the given points (-1,5), (0,3), and (3,9), let's substitute each point to the equation.
At (-1,5):
[tex]\text{ y = ax}^2\text{ + bx + c }\rightarrow5=a(-1)^2\text{ + b(-1) + c}[/tex][tex]\text{ 5 = a - b + c}[/tex]At (0,3):
[tex]\text{ y = ax}^2\text{ + bx + c }\rightarrow3=a(0)^2\text{ + b(0) + c}[/tex][tex]\text{ 3 = c}[/tex]At (3,9):
[tex]\text{ y = ax}^2\text{ + bx + c }\rightarrow9=a(3)^2\text{ + b(3) + c}[/tex][tex]\text{ 9 = 9a + 3b + c}[/tex]We now get these equations:
5 = a - b + c ; 3 =c; 9 = 9a + 3b + c
Let's determine the value of a, b and c. We get,
Substituting 3 = c to 5 = a - b + c,
[tex]\text{ 5 = a - b + c }\rightarrow\text{ 5 = a - b + 3 }\rightarrow\text{ a - b = 2}[/tex][tex]\text{ b = a - 2}[/tex]Let's substitute 3 = c and b = a - 2 to 9 = 9a + 3b + c,
[tex]\text{ 9 = 9a + 3b + c }\rightarrow\text{ 9 = 9a + 3(a-2) + 3}[/tex][tex]\text{ 9 = 9a + 3a - 6 + 3 }\rightarrow\text{ 12a = 9 + 6 - 3 }\rightarrow\text{ 12a = 12}[/tex][tex]\text{ a = }\frac{12}{12}\text{ = 1}[/tex]Since a = 1, let's solve for the value of b which is b = a - 2.
[tex]\text{ b = a - 2 }\rightarrow\text{ b = 1 - 2}[/tex][tex]\text{ b = -1}[/tex]Since we've identified that a = 1, b = -1 and c = 3, let's substitute the values to the standard form of a quadratic function to be able to make the equation.
[tex]\text{ y = ax}^2\text{ + bx + c }\rightarrow y=(1)x^2\text{ + (-1)x + (3)}[/tex][tex]\text{ y = x}^2\text{ - x + 3}[/tex]Therefore, the quadratic function in a standard form whose graph passes through the given points (-1,5), (0,3), (3,9) is y = x^2 - x + 3.
triangles FIM and LAK below are similar with m
8
Explanation
as the triangles are similar we can set a proportion
Step 1
Let
[tex]\text{ratio}=\frac{\text{longest side}}{\text{smallest side}}[/tex]so
a) for triangle FIM
[tex]\begin{gathered} \text{ratio}=\frac{\text{longest side}}{\text{middle side}} \\ ratio_1=\frac{FM}{FI}=\frac{6}{4}=\frac{3}{2} \\ ratio_1=\frac{3}{2} \end{gathered}[/tex]b) for triangle LAK
[tex]\begin{gathered} \text{ratio}=\frac{\text{longest side}}{\text{smallest side}} \\ ratio_2=\frac{LK}{LA}=\frac{12}{LA} \\ ratio_2=\frac{12}{LA} \end{gathered}[/tex]as the tringles are similar, the ratios are similar
hence
[tex]\begin{gathered} \text{ratio}_1=ratio_2 \\ \frac{3}{2}=\frac{12}{LA} \end{gathered}[/tex]Step 2
now, solve for LA
[tex]\begin{gathered} \frac{3}{2}=\frac{12}{LA} \\ \text{cross multiply } \\ 3\cdot LA=12\cdot2 \\ 3LA=24 \\ \text{divide both sides by 3} \\ \frac{3LA}{3}=\frac{24}{3} \\ LA=8 \end{gathered}[/tex]therefore, the answer i
8
I hope this helps you
At the lake, two companies are giving boat rides. At one booth a boat leaves every 12 minutes and at the
other booth a boat leaves every 18 minutes. In how many minutes will both boats be leaving at the same
time?
A) 6 minutes
B) 24 minutes
C) 36 minutes
C) 72 minutes
Answer:
Step-by-step explanation:B
Determine the distance between the two points (-1,-9) and (4,-7)What is the midpoint of the line segment joining the pairs of Points.
The distance between two points (x₁,y₁) and (x₂,y₂) is given by the following formula.
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Then, we have:
[tex]\begin{gathered} (x_1,y_1)=(-1,-9) \\ (x_2,y_2)=(4,-7) \end{gathered}[/tex][tex]\begin{gathered} d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \\ d=(4-(-1))^2+(-7-(-9))^2 \\ d=(4+1)^2+(-7+9)^2 \\ d=\sqrt{5^2+2^2} \\ d=\sqrt{25+4} \\ d=\sqrt{29} \\ d\approx5.4 \\ \text{ The symbol }\approx\text{ is read 'approximately'.} \end{gathered}[/tex]Finding the midpoint of the line segment joining the pointsThe midpoint of the line segment P(x₁,y₁) to Q(x₂,y₂) is:
[tex]\text{ Midpoint }=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]Then, we have:
[tex]\begin{gathered} \text{ Midpoint }=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) \\ \text{ Midpoint }=(\frac{-1+4}{2},\frac{-9+(-7)}{2}) \\ \text{ Midpoint }=(\frac{3}{2},\frac{-16}{2}) \\ \text{ Midpoint }=(\frac{3}{2},-8) \end{gathered}[/tex]AnswerThe distance between the given points is √29 units or 5.4 units rounded to the nearest tenth.
The midpoint of the line segment that joins the pairs of points is (3/2,-8).
The weight of a bacterium is defined by multiplying the functions of f(x) and g(x). Given f(x) = 6x6 + 8x and g(x) = 2x. Which of the following represents the weight of the bacterium?12x6 + 16x12x7 + 16x2-12x7 - 16x23x5 + 4
Given the following functions:
[tex]\begin{gathered} f(x)=6x^6+8x \\ g(x)=2x \end{gathered}[/tex]The weight of a bacterium is defined by multiplying the functions of f(x) and g(x).
So, the product of the functions will be as follows:
[tex]f(x)*g(x)=(6x^6+8x)*2x[/tex]We will use the distributive property to find the result as follows:
[tex]\begin{gathered} f(x)*g(x)=6x^6*2x+8x*2x \\ f(x)*g(x)=12x^7+16x^2 \end{gathered}[/tex]So, the answer will be 12x⁷+16x²
The common ratio for the home prices in an Austin neighborhood is 1.08 every year for the past 5 years increasing or decreasing? and is it linear or exponential? therefore interpret the change?
Lindsey, this is the solution:
Ratio for the home prices in an Austin neighborhood = 1.08 every year
1. It is an increasing ratio because it is higher than 1.
2. It is linear because the rate of change is constant (1.08)
3. Interpretation : The common ratio means that every year for the past 5 years the home prices in the Austin neighborhood grew 8%.
E3.This table shows the times, in minutes, It took 40 sixth grade students to run 1 mile.frequency15time (minutes)4 to less than 66 to less than 88 to less than 1010 to less than 1212 to less than 1414 to less than 16131272INTLDraw a histogram for the information in the table.321
The histogram is shown below:
A store is selling scooter for $40. You have coupon and purchase it for $15. What percentage was the coupon?
We can solve this problem by applying the rule of three:
[tex]\begin{gathered} 40\text{ dollars ------100\%} \\ 15\text{ dollars ------ x} \end{gathered}[/tex]hence,
[tex]x=\frac{(15)(100)}{40}[/tex]and it yields
[tex]x=\frac{1500}{40}[/tex]which result in x= 37.5. It means that 15 dollars corresponds to 37.5%
Find the slope of the line that goes through the points (2,-6) and (11,15).Slope,m=___Enter your answer as an integer or a reduced fraction in the form A/B
Answer:
m = 7/3
Explanation:
The slope of a line that passes through two points (x1, y1) and (x2, y2) can be calculated as
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]So, replacing (x1, y1) = (2, -6) and (x2, y2) = (11, 15), we get:
[tex]m=\frac{15-(-6)}{11-2}=\frac{15+6}{9}=\frac{21}{9}=\frac{7}{3}[/tex]Therefore, the slope is 7/3
Compare the quantities in Column A and Column B Column A Column B The solutions of 4x - 30 2-3x + 12 The solutions (A) The quantity in Column A is greater. (B) The quantity is Column B is greater. (C) The quantities are equal. (D) The relationship cannot be determined from the inform
Column A:
[tex]4x-30\ge-3x+12[/tex]The solution will be as following :
[tex]\begin{gathered} 4x+3x\ge12+30 \\ 7x\ge42 \\ \frac{7x}{7}\ge\frac{42}{7} \\ \\ x\ge6 \end{gathered}[/tex]Column B:
[tex]\frac{1}{2}x+3<-2x-6[/tex]The solution will be as following :
[tex]\begin{gathered} \frac{1}{2}x+2x<-6-3 \\ 2\frac{1}{2}x<-9 \\ \frac{5}{2}x<-9 \\ \\ x<-9\cdot\frac{2}{5} \\ \\ x<-3.6 \end{gathered}[/tex]Compare the quantities in Column A and Column B
so,
[tex]x\ge6\text{ and x < -3.6}[/tex]So, the answer is option A) The quantity in Column A is greater.
HELPPP please
Which of the following products is irrational?
Answer: B) 7 x π
Step-by-step explanation:
π is irrational since it doesn't have an end(yet)
This is a one step inequality can you help my find the answer I don't know how to do this X + 7 < 19
Answer:
X<12
Step-by-step explanation:
A school is planning a 4th grade field trip. There are 157 students and 9 teachers in the 4th gradeIf each bus holds 45 people, how many buses does the school need to make the field trip?Which of the following equations can be used to solve this problem?
Given:
A school is planning a 4th grade field trip. There are 157 students and 9 teachers in the 4th grade. Each bus holds 45 people.
Required:
To find the number of buses does the school need to make the field trip.
Final Answer:
There area total
[tex]\begin{gathered} =157+9 \\ =166 \end{gathered}[/tex]166 people.
Let the number of bus be x.
Each bus holds 45 people, therefore
[tex]\begin{gathered} 45x=166 \\ x=\frac{166}{45} \\ x=3.68 \\ x\approx4 \end{gathered}[/tex]Final Answer:
4 buses need to make the field trip.
Which value of y makes the equation true?
-2y-9=-11
Answer:
Y=1
Step-by-step explanation:
To solve this we can start by isolating the y value.
To get rid of the -9 we add nine to both sides leaving us with
-2y=-2
Then we want to only have y = ?
So we divide -2 by -2 and get one
y=1
i need help with this!
Answer:
19. 7
20. 11
21. -12
22. 15
23. 77
24. −6
Step-by-step explanation:
Simply plug the values of a, b and c into each equation and evaluate using a calculator or manually
QUESTION IN SCREENSHOT, FIRST PERSON MARKING BRANLIEST
The equation of a line in slope intercept form parallel to 6x + 5y = 11 and passes through (-2, -8) is y = (-6/5)x - (28/5)
What is an equation?An equation is an expression that can be used to show the relationship between numbers and variables.
The equation of a line in slope intercept form is:
y = mx + b
Where m is the slope and b is the y intercept
Two lines are parallel if they have same slope
Given a line:
6x + 5y = 11
5y = -6x + 11
y = (-6/5)x + (11/5)
The line parallel to 6x + 5y = 11 have a slope of -6/5. The line passes through (-2, -8), hence:
y - (-8) = (-6/5)(x - (-2))
y + 8 = (-6/5)x + 12/5
y = (-6/5)x - (28/5)
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1)What Miller is adding a room to the back of his house. For the foundation, 12 ft wide, & and 4 ft deep. How many cubic ft of soil have to be Removed?
We need to find the volume, the volume can be found as:
[tex]V=w\cdot l\cdot h[/tex]Where:
w = width = 12ft
l = length = 16ft
h = height = 4ft
so:
[tex]\begin{gathered} V=12\cdot16\cdot4 \\ V=768ft^3 \end{gathered}[/tex]He has to remove 768ft³ of soil
Suppose the First Bank of Lending offers a CD (Certificate of Deposit) that has a 6.45% interest rate andis compounded quarterly for 3 years. You decide to invest $5500 into this CD.a) Determine how much money you will have at the end of three years.b) Find the APY.
In order to solve this, we have to use the compound interest formula given by the following expression:
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]Where r is the interest rate, P is the initial amount deposited, n the number of times the period is compounded a year, t the year, and A the final amount.
By replacing 0.0645 (6.45%) for r, 4 for n, 3 for t and 5500 for P into the above equation, we get:
[tex]A=5500(1+\frac{0.0645}{4})^{4\times3}=6663.8978[/tex]Then, after 3 years you will have $6663.9.
In order to determine the APY, we can use the following formula:
[tex]APY=100\times((1+r/n)^n-1)[/tex]Where n is the number of times the interest is compounded a year (4) and r is the rate of interest (0.0645), then we get:
[tex]APY=100\times((1+0.0645\/4)^4-1)=6.61[/tex]Then, the APY equals 6.61%
which number is a solution of the inequality 8 - 1/4 b > 27
The inequality is 8-1/4x>27. The solution of the inequality is b<-76.
Given that,
The inequality is 8-1/4x>27
We must determine how to address the inequity.
Take,
8-1/4x>27
Multiply the inequality's two sides by its lowest common denominator,
4×8-4×1/4b>27×4
Reduce the expression to the lowers term,
4×8-b>4×27
Calculate the product or quotient,
32-b>4×27
Calculate the product or quotient,
32-b>108
Rearrange unknown terms to the left side of the equation,
-b>108-32
Calculate the sum or difference,
-b>76
Divide the inequality's two sides by the variable's coefficient,
b<-76
Therefore, the solution of the inequality is b<-76.
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Leila deposits the same amount of money into a bank account every month. The table below shows the amount of money in the account after different amounts of time.
To see how money is changing with respect to time, we will observe the time and money value differences between two periods.
At 6 months, there is $467
At 8 months, there is $557
We can see that within a two month increase, the amount of money has also increased.
We can observe the values for 10 and 12 months and see that these months are also asscociated with increased account values.
a)
Correct option: As the time increases the amount of money in the account increases.
Rate of increase:
r = (557-467)/2 =
$45 dollars per month.
We are asked to find the amount of money at time t= 0 months.
Since then, the amount in the account has increased 6 times. It has increased by $270.
b)
Therefore the account started with $197.
Maple tree diameters in a forest area are normally distributed with mean 10 inches and standard deviation 2.2 inches. Find the proportion of trees having a diameter greater than 15 inches.
Given:
[tex]\begin{gathered} \mu=10\text{ }inches \\ \sigma=2.2\text{ inches} \end{gathered}[/tex]To find- P(X>15)
Explanation-
We know that a z-score is given by-
[tex]z=\frac{x-\mu}{\sigma}[/tex]where x is the raw score, mu is the mean and sigma is the standard deviation.
Hence, the proportion of trees having a diameter greater than 15 inches will be-
[tex]\begin{gathered} P(x>15)=P(\frac{x-\mu}{\sigma}>\frac{15-\mu}{\sigma}) \\ P(x>15)=P(Z>\frac{15-10}{2.2}) \end{gathered}[/tex]On further solving, we get
[tex]\begin{gathered} P(x\gt15)=P(Z\gt\frac{5}{2.2}) \\ P(x\gt15)=P(Z\gt2.2727) \end{gathered}[/tex]With the help of an online tool, the probability will be
[tex]P(x>15)=0.0115[/tex]Since the significance level is not mentioned, we assumed it is 0.05.
Thus, the proportion of trees having a diameter greater than 15 inches is 0.0115.
The answer is 0.0115.
At a particular restaurant, each slider bas 200 calories and each mini hotdog bas 100calories. A combination meal with mini hotdogs and sliders is shown to have 1200total calories and 4 times as many mini botdogs as there are sliders. Graphically solveagystem of equations in order to determine the number of sliders in the combinationmeal, 2, and the number of mini hotdogs in the combination meal, y.
x: the number of sliders in the combination meal
y: the number of mini hotdogs in the combination meal
Each slider has 200 calories and each mini hotdog has 100 calories. A combination meal with mini hotdogs and sliders is shown to have 1200
total calories, means:
200x + 100y = 1200
The combination meal has 4 times as many mini hotdogs as there are sliders, means:
y = 4x
Lucius has at most $80 to spend on clothes. He wants to buy a pair of jeans for $22 and spend the rest on t-shirts. Each T-shirt costs $15. How many shirts can Lucius buy?
Answer:
He can only buy 3 t-shirts
Step-by-step explanation:
80-22= 58
58-(15*3) = 13
A fitness club offers two water aerobics classes. There are currently 40 people in the moming class and
attendance is growing at a rate of 2 people per month. The afternoon class has 22 members and is growing at
a rate of 8 people per month. In how many months will there be the same number of people in each class and
how many people will be in each class?
please help
Answer:
3 months
Step-by-step explanation:
You would set the equations = to each other to identify when they will be the same number of __. The equations to begin with is 40 + 2x and 22 + 8x, you would do 40 + 2x = 22 + 8x and algebraically solve for X, which is the months. so 18 = 6x, x = 3
A pulley is turning at an angular velocity of 14.0 rad per second. How many revolutions is the pulley making each second? (Hint: one revolution equals 2 pi rad)
Answer:
7/π ≈ 2.23 revolutions per second
Step-by-step explanation:
You want the know the angular velocity in revolutions per second of a pulley turning at 14.0 radians per second.
Unit ConversionThe velocity in rad/s can be converted to rev/s using the conversion factor ...
1 rev = 2π rad
The angular velocity is ...
[tex]\dfrac{14\text{ rad}}{\text{s}}\times\dfrac{1\text{ rev}}{2\pi\text{ rad}}=\dfrac{14}{2\pi}\,\dfrac{\text{rev}}{\text{s}}=\boxed{\dfrac{7}{\pi}\text{ rev/s}\approx2.23\text{ rev/s}}[/tex]
. Amy's school is selling
tickets to a choral
performance. A senior
citizen's ticket is $6 and a
child's ticket is $15. If they
made $810 dollars and
sold a total of 72 child
and senior citizen tickets,
how many of each ticket
did they sell?
By solving the equation we know that Amy's school sold 30 tickets to seniors and 42 tickets to children.
What are equations?A mathematical statement called an equation includes the symbol "equal to" between two expressions with equal values. Consider the formula 3x + 5 = 15. Different types of equations exist, including linear, quadratic, cubic, and others. Any value of the variable that satisfies the equality, that is, makes the Left Hand Side (LHS) and the Right Hand Side (RHS) of the equation equal, is a solution of the equation. Finding an equation's solution or solution is known as solving the equation.So, let 's' represents seniors and 'c' represents children.
The equation is as follows:
6s + 15c = 810 - 6(s + c = 72)⇒ -6s - 6c = -432 (Cut s)Then,
9c = 378c = 378/9c = 42Then,
s = 72 - 42s = 30Therefore, by solving the equation we know that Amy's school sold 30 tickets to seniors and 42 tickets to children.
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If R(-2,-1) is the midpoint of ST and S(-14,3),find the coordinates of t
Answer
Explanation
Mathematically, if a point R(x, y) divides the coordinates S (x₁, y₁) and T (x₂, y₂) internally in the ratio m:n then point R(x, y) is given as
x = [(mx₂ + nx₁)/(m + n)]
y = [(my₂ + ny₁)/(m + n)]
For this question, we are given that
R (x, y) = R(-2, -1)
S (x₁, y₁) = S (-14, 3)
T (x₂, y₂) = ?
Since it is divided equally into two parts (As per the midpoint), m : n = 1 : 1
x = -2
y = -1
x₁ = -14
y₁ = 3
x₂ = ?
y₂ = ?
m = 1
n = 1
x = [(mx₂ + nx₁)/(m + n)]
-2 = [(1 × x₂) + (1 × -14)]/(1 + 1)
-2 = [x₂ - 14]/2
[tex]\begin{gathered} -2=\frac{x_{2}-14}{2} \\ \text{Cross multiply} \\ x_{2}-14=2\times-2 \\ x_{2}-14=-4 \\ x_{2}=-4+14 \\ x_{2}=10 \end{gathered}[/tex]y = [(my₂ + ny₁)/(m + n)]
-1 = {
Simplify the expression by combining the radical terms using the indicated operations(s) Assume all variables are positive.
Answer:
[tex]38x\sqrt[]{34xy}[/tex]Step-by-step Explanation:
Given the below expression;
[tex]8x\sqrt[]{34xy}+3x\sqrt[]{34xy}+9x\sqrt[]{306xy}[/tex]We'll go ahead and simplify the given expression following the below steps;
Step 1: Combine like terms;
[tex]\begin{gathered} (8x\sqrt[]{34xy}+3x\sqrt[]{34xy})+9x\sqrt[]{306xy} \\ 11x\sqrt[]{34xy}+9x\sqrt[]{306xy} \end{gathered}[/tex]Step 2: Split the radicand of the second term as seen below;
[tex]\begin{gathered} 11x\sqrt[]{34xy}+9x\sqrt[]{9\cdot34\cdot xy} \\ =11x\sqrt[]{34xy}+9x(\sqrt[]{9}\cdot\sqrt[]{34xy}) \\ =11x\sqrt[]{34xy}+9x\cdot3\sqrt[]{34xy} \\ =11x\sqrt[]{34xy}+27x\sqrt[]{34xy} \end{gathered}[/tex]
Step 3: Combine like terms;
[tex]\begin{gathered} 11x\sqrt[]{34xy}+27x\sqrt[]{34xy} \\ =38x\sqrt[]{34xy} \end{gathered}[/tex]
Simplify the complex rational expression by the method of your choice. 1——x-6———-1 - 1 —- x-6
To find:
The simplified form of the rational expression.
Solution:
The given rational expression can be simplified as follows:
[tex]\begin{gathered} \frac{\frac{1}{x-6}}{1-\frac{1}{x-6}}=\frac{\frac{1}{x-6}}{\frac{x-6-1}{x-6}} \\ =\frac{1(x-6)}{(x-6)(x-7)} \\ =\frac{1}{x-7} \end{gathered}[/tex]Thus, the answer is:
[tex]\frac{1}{x-7}[/tex]Give two systems of equations that would be easier to solve by substitution than by elimination. Then give two systems that would be easier to solve with elimination. Finally, explain how you decide whether to use elimination or substitution to solve a system.
Please don't answer too complicated
Step-by-step explanation:
if the given equations are linear, then no matter which method is used, it depends on pupils ability/habbits, but usually 'by elimination' is easier, then 'by substitution';
in the most cases the 'by substitution' can be used only (systems of non-linear equations).
Example 1. This system can be solved by any method, but 'by elimination' is shorter:
[tex]\left \{ {{x+y=2} \atop {x-y=2}} \right.[/tex]
Example 2. This system can be solved by any method, but 'by substitution' is shorter:
[tex]\left \{ {{2x+y=3} \atop {7x+3y=10}} \right.[/tex]
A company makes pens. They sell each pen for $ 6
Answer:
a. -3,000
b. 1,750
Explanation:
We were given the following information:
A company makes pens:
Each pen is sold at $6 per unit
Revenue = 6 * x
Manufacture Cost = 2 * x
Start-up Cost = $7,000
Cost = Manufacture Cost + Start-up Cost = 2 * x + 7,000
Profit = Revenue - Cost
a) The profit is calculated for 1,000 pens as shown below:
[tex]\begin{gathered} Profit=Revenue-Cost \\ \text{For the making of 1,000 pens, it means: }x=1,000 \\ Revenue=6\cdot x=6\times1,000 \\ Revenue=\text{\$}6,000 \\ Cost=2\cdot x+7,000 \\ Cost=2\times1,000+7,000 \\ Cost=2,000+7,000 \\ Cost=\text{\$}9,000 \\ \\ Profit=6,000-9,000 \\ Profit=-\text{\$}3,000 \\ \\ \therefore Profit=-\text{\$}3,000 \end{gathered}[/tex]Hence, the profit is -3,000
b) We will calculate for the number of pens needed to be sold for the company to break even as shown below. We have:
[tex]\begin{gathered} \text{At breakeven: }Revenue=Cost \\ \Rightarrow6x=2x+7,000 \\ \text{We will calculate for the value of the variable ''x'':} \\ 6x=2x+7,000 \\ \text{Subtract ''2x'' from both sides, we have:} \\ 6x-2x=7,000 \\ 4x=7,000 \\ \text{Divide both sides by ''4'', we have:} \\ x=\frac{7,000}{4} \\ x=1,750 \\ \\ \therefore x=1,750 \end{gathered}[/tex]Hence, the breakeven occurs when the company has made 1,750 pens
